text: After entering the continental shelf, a tidal wave quickly faces a boundary in the form of a landmass. When the tidal wave reaches a continental margin, it continues as a boundary trapped Kelvin wave. Along the coast, a boundary trapped Kelvin is also known as a coastal Kelvin wave or Edge wave. A Kelvin wave is a special type of gravity wave that can exist when there is (1) gravity and stable stratification, (2) sufficient Coriolis force and (3) the presence of a vertical boundary. Kelvin waves are important in the ocean and shelf seas, they form a balance between inertia, the Coriolis force and the pressure gradient force. The simplest equations that describe the dynamics of Kelvin waves are the linearized shallow water equations for homogeneous, in-viscid flows. These equations can be linearized for a small Rossby number, no frictional forces and under the assumption that the wave height is small compared to the water depth (). The linearized depth-averaged shallow water equations become: 
  u momentum equation: 
  v momentum equation: 
  the continuity equation:
  where  is the zonal velocity ( direction),  the meridional velocity ( direction),  is time and  is the Coriolis frequency.
  Kelvin waves are named after Lord Kelvin, who first described them after finding solutions to the linearized shallow water equations with the boundary condition . When this assumption is made the linearized depth-averaged shallow water equations that can describe a Kelvin wave become: 
  u momentum equation:
  v momentum equation:
  the continuity equation:
  Now it is possible to get an expression for , by taking the time derivative of the continuity equation and substituting the momentum equation:
  The same can be done for , by taking the time derivative of the v momentum equation and substituting the continuity equation
  Both of these equations take the form of the classical wave equation, where . Which is the same velocity as the tidal wave and thus of a shallow water wave. These preceding equations govern the dynamics of a one-dimensional non-dispersive wave, for which the following general solution exist:
  where length  is the Rossby radius of deformation and  is an arbitrary function describing the wave motion. In the most simple form  is a cosine or sine function which describes a wave motion in the positive and negative direction. The Rossby radius of deformation is a typical length scale in the ocean and atmosphere that indicates when rotational effects become important. The Rossby radius of deformation is a measure for the trapping distance of a coastal Kelvin wave. The exponential term results in an amplitude that decays away from the coast. 
  The expression of tides as a bounded Kelvin wave is well observable in enclosed shelf seas around the world (e.g. the English channel, the North Sea or the Yellow sea). Animation 1 shows the behaviour of a simplified case of a Kelvin wave in an enclosed shelf sea for the case with (lower panel) and without friction (upper panel). The shape of an enclosed shelf sea is represented as a simple rectangular domain in the Northern Hemisphere which is open on the left hand side and closed on the right hand side. The tidal wave, a Kelvin wave, enters the domain in the lower left corner and travels to the right with the coast on its right. The sea surface height (SSH, left panels of animation 1), the tidal elevation, is maximum at the coast and decreases towards the centre of the domain. The tidal currents (right panels of animation 1) are in the direction of wave propagation under the crest and in the opposite direction under the through. They are both maximum under the crest and the trough of the waves and decrease towards the centre. This was expected as the equations for  and  are in phase as they both depend on the same arbitrary function describing the wave motion and exponential decay term.  Therefore this set of equations describes a wave that travels along the coast with a maximum amplitude at the coast which declines towards the ocean. These solutions also indicate that a Kelvin wave always travels with the coast on their right hand side in the Northern Hemisphere and with the coast at their left hand side in the Southern Hemisphere. In the limit of no rotation where , the exponential term increase without a bound and the wave will become a simple gravity wave orientated perpendicular to the coast. In the next section, it will be shown how these Kelvin waves behaves when traveling along a coast, in an enclosed shelf seas or in estuaries and basins.